A287060 Primes a(n) such that a(1) = 2, a(2) = 3, and a(n) is the smallest prime greater than a(n-1) such that (a(n) - 1)/2 is not divisible by a(m) for all m < n.

2, 3, 11, 47, 59, 71, 83, 107, 131, 179, 191, 227, 239, 251, 311, 347, 431, 443, 467, 491, 563, 587, 599, 647, 719, 743, 839, 911, 971, 1019, 1031, 1091, 1103, 1151, 1187, 1259, 1283, 1307, 1319, 1367, 1427, 1451, 1511, 1523, 1559, 1571, 1583, 1619, 1667

Offset: 1

Author

Amiram Eldar, May 19 2017

Keywords

• nonn

Comments

As in A100564, the number of terms in this sequence which do not exceed x is ~ (1 + o(1)) x/(logx loglogx), thus the sum of the their reciprocals diverges.

Examples

			5 and 7 are not in the sequence since a(1) | (5 - 1)/2 and a(2) | (7 - 1)/2.
a(3) = 11 is in the sequence since (11 - 1)/2 = 5 is not divisible by 2 or 3.
		

References

• Jean-Marie De Koninck and Florian Luca, Analytic Number Theory: Exploring the Anatomy of Integers, American Mathematical Society, 2012, Problem 15.1, p. 263.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A100564.

Programs

Mathematica

a[1] = 2; a[2] = 3; a[n_] := a[n] = Block[{k=PrimePi[a[n - 1]] + 1, t=Table[a[i], {i, n-1}]}, While[Union[ Mod[(Prime[k] - 1)/2, t]][[1]] == 0, k++]; Prime[k]]; Table[a[n], {n, 49}]

PARI

isok(p, va) = {q = (p-1)/2; for (k=1, #va, if (!(q % va[k]), return (0));); return (1);}
lista(nn) = {va = [2, 3]; print1(va[1], ", " va[2], ", "); for (n=3, nn, forprime(p=nextprime(vecmax(va)+1),, if (isok(p, va), va = concat(va, p); print1(p, ", "); break);););} \\ Michel Marcus, May 21 2017